Principios del Cooperativismo:
NIVEL LEGISLATIVO
As will be shown below, the magnetic anisotropy of a fully stretched PEG-PS block copolymer, ∆χP (see figure 3.4), is negative, meaning that according to equation 2.7, it aligns perpendicular to an applied magnetic field. However, as we will see, the magnetic energy of one single PEG-PS polymer is not large enough to compete with thermal fluctuations, so an aggregate or an assembly of PEG-PS polymers is necessary to obtain magnetic alignment. The following initial calculation presented is only meant to give insight into the sign of the magnetic anisotropy for fully extended polymers. In that case, the magnetic anisotropy of a single PEG-PS block-copolymer, ∆χP, can be written in terms of the average magnetic anisotropy of a PEG or PS monomer as follows:
∆χP= n · ∆χPEG+ m · ∆χPS, (3.43) with n the number of repeating units of PEG and m the number of repeating units of PS. In first instance, the anisotropy of the magnetic susceptibility ∆χ is calculated for fully extended PEG and PS polymers. Since in reality the polymers in the membrane are coiled to some extent, the absolute value of the magnetic anisotropy will be lower than the value calculated here, which represents the maximal ∆χ possible. The effect of coiling will be considered later.
Diamagnetic anisotropy of PEG
In Figure 3.5a, one monomer of PEG is shown. This monomer consists of one C-O-C and one C-C group as shown in Figure 3.5b and 3.5c1. The diamagnetic anisotropies of these groups (in 10−12m3/mol) are [15]:
χCOCi = χCOCi ,
χCOCj = χCOCi + ∆χCOCij = χCOCi + 49, (3.44) χCOCk = χCOCi + ∆χCOCik = χCOCi + 82,
1
The contributions of the C-H bonds are being neglected since they are much smaller than those of C-C and C-O bonds
3.3 Magnetic properties
and [14]:
χCCi = χCCi ,
χCCj = χCCi + ∆χCCij = χCCi + 16, (3.45) χCCk = χCCi + ∆χCCik = χCCi + 16.
The C-O-C frame (i,j,k) coincides with the lab frame of the polymer (x, y, z) and therefore:
χCOCx = χCOCi ,
χCOCy = χCOCj , (3.46) χCOCz = χCOCk .
For simplicity, we shall assume that all backbone atoms are bonded to their neighboring atoms by perfect tetrahedral symmetry. In that case, the i-axis of the C-C frame (i,j,k) makes an angle of 35.5◦ with respect to the x-axis of the polymer frame (x, y, z) and therefore:
χCCx = χCCi cos2(35.5) + χCCj sin2(35.5) = χCCi + 5.4,
χCCy = χCCi sin2(35.5) + χCCj cos2(35.5) = χCCi + 10.6, (3.47) χCCz = χCCk = χCCi + 16.
Figure 3.4: A PEG-PS block copolymer in a magnetic field B. The magnetic susceptibility along the polymer chain is smaller than the magnetic suscepti- bility perpendicular to it, leading to a negative ∆χP.
3 Calculations on polymers and polymer vesicles
This gives a total diamagnetic susceptibility per monomer of PEG of:
χPEGx = χCOCi + χCCi + 5.4,
χPEGy = χCOCj + χCCj + 59.6, (3.48) χPEGz = χCOCk + χCCk + 98.
Assuming that the polymers in the membrane are rotated along the molecular x-axis in random orientations, χy and χz will average out to one component
perpendicular to χx. This leads to the following magnetic anisotropy:
∆χPEG= χPEGx −χ
PEG
y + χPEGz
2 = −73.4. (3.49)
Figure 3.5: (a) One monomer of PEG in the lab frame. (b) C-O-C group in
the molecular frame in which χCOC is defined. (c) C-C group in the molecular
frame in which χCCis defined. (d) one monomer of PS in the lab frame showing
the angle that the backbone C-C’s make with respect to the x-axis. (e) One monomer of PS in the lab frame showing the angle of the C-C connecting the phenyl to the backbone with respect to the y-axis. (f) a phenyl group in the
molecular frame in which χPh is defined. (g) the phenyl is assumed to rotate
3.3 Magnetic properties
So the magnetic anisotropy of one monomer of PEG when the polymer is fully extended is ∆χPEG = −73.4 · 10−12 m3/mol.
Diamagnetic anisotropy of PS
In Figure 3.5d, one repeating unit of polystyrene is shown in the lab frame. It consists of one phenyl group, two C-C bonds in the backbone and one C-C bond connecting the phenyl to the backbone.
We will consider the C-C bonds in the backbone first. The i-axis of the C-C frame (i,j,k) makes an angle of 35.5◦ with respect to the x-axis of the polymer frame (x, y, z) for both C-C bonds in the backbone. The contribution of each of these two bonds to the magnetic susceptibility of the PS monomer in the frame of the polymer can be written as:
χCCx = χCCi + 5.4,
χCCy = χCCi + 10.6, (3.50) χCCz = χCCi + 16.
Now, we will consider the C-C bond connecting the phenyl to the backbone (Figure 3.5f). The i-axis of the C-C frame (i,k,j) makes an angle of 54.5◦ (half of the angle between two bonds in a SP3 hybridized molecule, which is 109◦) with respect to the y-axis of the polymer frame (x, y, z) and therefore:
χCCx = χCCi + 16,
χCCy = χCCi cos2(54.5) + χCCj sin2(54.5) = χCCi + 10.6, (3.51) χCCz = χCCi sin2(54.5) + χCCj cos2(54.5) = χCCi + 5.4.
The three C-C bonds in a PS monomer thus give:
χallCCx = 3χCCi + 26.8,
χallCCy = 3χCCi + 31.8, (3.52) χallCCz = 3χCCi + 37.4.
Assuming that the polymers in the membrane are rotated along the molecular x-axis in random orientations, χy and χz will average out to one component
which is perpendicular to χx. This leads to the following magnetic anisotropy:
∆χallCC= χallCCx −χ
allCC
y + χallCCz
3 Calculations on polymers and polymer vesicles
Finally, the contribution of the phenyl group needs to be considered (Figure 3.5e). The diamagnetic susceptibility of a phenyl group (in 10−12 m3/mol) is:
χPhi = −439,
χPhj = −439, (3.54) χPhk = −1189,
with the molecular axis i, j, k as defined in Figure 3.5e. The i-axis of the phenyl frame makes an angle of 54.5◦ with respect to the y-axis of the polymer frame (Figure 3.5f). Also, the phenyl group can rotate along the C-C bond that connects it to the backbone, thereby introducing an angle φ (with φ = 0 corresponding to the configuration as drawn in Figure 3.5g). The diamagnetic susceptibility of the phenyl group in the polymer frame can therefore be written as:
χPhx = χPhk cos2(φ) + χPhj sin2(φ) ,
χPhy =χPhk sin2(φ) + χPhj cos2(φ)sin2(θ) + χPhi cos2(θ) , (3.55) χPhz =
χPhk sin2(φ) + χPhj cos2(φ)
cos2(θ) + χPhi sin2(θ) .
In Figure 3.6, χPhx , χPhy , χPhz and χPhy,z, the average of χPhy and χPhz , are plot- ted as function of φ. χPh
x is smaller than χPhy,zfor angles of 0◦ to 54.7◦and 125.3◦
to 180◦. This means that in this region the contribution of the phenyl groups leads to a perpendicular alignment of the PS relative to the magnetic field. The angle φ (or possible angles φ) that the phenyl group can adopt determines the diamagnetic susceptibilities in the x, y and z directions. If the phenyl group is free to rotate, or if many phenyl rings adopt all possible orientations equally, the average contribution to the anisotropy of the diamagnetic susceptibility per phenyl is: ∆χPh= −188.3 · 10−12m3/mol. A freely rotating phenyl group leads to a ∆χPhx which is lower than ∆χPhy,z which supports the idea that the polymer aligns perpendicularly to the magnetic field. Adding to this the contribution of the backbone, we get: ∆χPS= −196.1 · 10−12 m3/mol.
The calculated ∆χ’s for PEG and PS are for maximally extended polymers, so they represent the maximum ∆χ possible. The phenyl groups as well as the PS and PEG backbones all contribute to a negative ∆χ, which means that the polymers will align perpendicular to an applied magnetic field. The magnetic anisotropy per repeating unit is about 3 times as large for PS than for PEG. Also, in a typical PEG-PS block copolymer the number of PS units is at least 3 times as large. This means that the contribution to the magnetic anisotropy is
3.3 Magnetic properties
Figure 3.6: The magnetic susceptibility of a phenyl group in the lab frame (x, y and z directions) as function of the angle φ.
dominated by the PS rather than the PEG. Also, in reality, the polymer chain will not be fully extended but rather be randomly coiled to some extent [16,17]. This will decrease the absolute value of ∆χ as will be discussed next.
The effect of coiling on the magnetic anisotropy of PS
In a polymersome membrane the block-copolymers are not fully stretched since this configuration is statistically improbable. Rather, the block-copolymer will be coiled to some unknown extent [16, 17]. Therefore, one should investigate how the magnetic anisotropy depends on the degree of coiling. For this purpose, a Matlab script was written that calculates the magnetic anisotropy for PS as function of the degree of coiling. As an input parameter, the fraction of maximal extension is given, which determines the projection of the backbone C-C bonds on the x-axis. Then, all possible directions in the y and z directions are calculated, given the angles for certain bonds. From these possibilities, one is randomly chosen. This has no effect on the outcome of ∆χPS, since the contributions in the y and z direction are averaged anyway. Again, the phenyl is allowed to rotate around the C-C bond connecting it to the backbone.
The result of the calculation is given in Figure 3.7a. It shows how the mag- netic anisotropy per repeating unit, ∆χPS depends on the degree of polymer
3 Calculations on polymers and polymer vesicles
Figure 3.7: (a) The magnetic anisotropy of PS (given per repeating unit) as function of extension. The largest contribution comes from the phenyl group. The phenyl is allowed to rotate freely around the C-C bond connecting it to the backbone. (b) Examples of differently extended polymers consisting of 24 PS units. The top one is fully extended while the bottom one is extended for only 73.3%. The one that is partially extended occupies more space in the yz plane. All distances are in nm.
3.3 Magnetic properties
extension. The contributions from the individual components are also given. The plot clearly shows that the magnetic anisotropy is dominated by the con- tribution of the phenyl group. This is mainly because the phenyl group has a much larger magnetic anisotropy than a C-C group. Furthermore, one can see that the contribution of the C-C bond connecting the phenyl to the backbone is of opposite sign compared to the contributions of the backbone C-C’s. This makes the contribution of all C-C bonds together even smaller.
If the polymer gets more extended, ∆χPS will become more negative. Coil-
ing will lead to a conformation in which the polymer occupies less space in the x-direction but more space in the yz-plane, as can be seen in Figure 3.7b-d. The effect is that ∆χPS becomes smaller upon coiling up to a point where it is
actually zero. Upon further reduction of the projection of the x-axis the sign of ∆χPS flips, meaning that the polymer becomes more extended in the yz-plane. In principle, this means that the polymer is stretched again, but this time in the yz-plane rather than in the x-direction.